The Nonlinear Schrodinger Equation for Orthonormal Functions II: Application to Lieb-Thirring Inequalities

In this paper we disprove part of a conjecture of Lieb and Thirring concerning the best constant in their eponymous inequality. We prove that the best Lieb-Thirring constant when the eigenvalues of a Schrodinger operator -Delta + V(x) are raised to the power. is never given by the one-bound state case when kappa > max(0, 2 - d/2) in space dimension d >= 1. When in addition kappa >= 1 we prove that this best constant is never attained for a potential having finitely many eigenvalues. The method to obtain the first result is to carefully compute the exponentially small interaction between two Gagliardo-Nirenberg optimisers placed far away. For the second result, we study the dual version of the Lieb-Thirring inequality, in the same spirit as in Part I of this work Gontier et al. (The nonlinear Schrodinger equation for orthonormal functions I. Existence of ground states. Arch. Rat. Mech. Anal, 2021. https://doi.org/10.1007/s00205-021-01634-7). In a different but related direction, we also show that the cubic nonlinear Schrodinger equation admits no orthonormal ground state in 1D, for more than one function.